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Luz Roncal

Publications and source records attributed to Luz Roncal.

At least 19 recordsLinked to original sources
arXiv2026

Stein's generalized spherical maximal operator

This survey concerns Stein's analytic family of generalized spherical means and the associated maximal operator. We review Stein's original theorem, the sharp necessary conditions for boundedness, and the complete theory for radial functions. We then explain the connection between positive results and local smoothing estimates for the half-wave propagator and, more generally, for Fourier integral operators. This point of view yields the sharp two-dimensional result through the work of Guth, H. Wang, and R. Zhang and, when combined with the recent local smoothing theorems, an improved higher-dimensional range. As a byproduct, the maximal operator associated with the first radial derivative of spherical means is bounded on $L^p(\R^4)$ for $2<p<3$, a case left open in recent work of Kwaśnicki. We keep track of the relation between the local smoothing exponents and the resulting maximal estimates, and identify the region that remains open.

math.CA↗
arXiv2026

Quantitative Landis-type result for Dirac operators

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $( \mathcal{D}_n + \mathbb{V} ) φ= 0$ satisfies a lower bound of order $\exp(-κR^{2} (\log R)^{2})$ as $|x|=R\to \infty$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain--Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local Hölder regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.

math.AP↗
arXiv2026

Landis' conjecture: a survey

We survey Kondrat'ev--Landis' conjecture, providing an up-to-date account of the main advances and describing the techniques developed. We complement the overview with references and formulations of the problem in further closely connected contexts.

math.AP↗
arXiv2026

The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$

In this paper, we study the bilinear cone multiplier operator in two dimensions. We establish $L^{p_1}\times L^{p_2}\to L^{p}$ boundedness for a regularized version of this operator over a broad range of exponents satisfying the Hölder scaling condition. Our approach is based on a decomposition of the bilinear operator into square functions associated with linear cone multipliers and their variants. We derive pointwise bounds for these square functions via suitable strong maximal function estimates, and obtain sharp $L^4$ bounds using geometric methods originating in the work of Córdoba and Carbery. The combination of these estimates yields the $L^p$ boundedness for the bilinear cone multiplier.

math.CA↗
arXiv2026

A lower bound for a variation norm operator associated with circular means

We prove that a local $L^p(V_2)$ variation norm estimate fails for circular means in two dimensions, and quantify this failure by proving lower bounds for functions of exponential type. This is related to lower bounds for Fourier multipliers supported on annuli, of the type considered by Córdoba.

math.CA↗
arXiv2025

Strichartz and local smoothing estimates for the fractional Schrödinger equations over fractal time

We obtain Strichartz-type estimates for the fractional Schrödinger operator $f \mapsto e^{it(-Δ)^{γ/2}} f$ over a time set $E$ of fractal dimension. To obtain those estimates capturing fractal nature of $E$, we employ the notions in the spirit of the Assouad dimension, such as, bounded Assouad characteristic and Assouad specturm. We also prove the estimate $$ \| e^{it(-Δ)^{γ/2}} f \|_{L_t^q(\mathrm{d}μ; L_x^r(\mathbb{R}^d))} \le C \|f\|_{H^s}, $$ where $μ$ is a measure satisfying an $α$-dimensional growth condition. In addition, we establish related inhomogeneous estimates and $L^2$ local smoothing estimates. A surprising feature of our work is that, despite dealing with rough fractal sets, we extend the known estimates for the fractional Schrödinger operators in a natural way, precisely consistent with the associated fractal dimensions.

math.AP↗
arXiv2025

Endpoint estimates for the fractal circular maximal function and related local smoothing

Sharp $L^p$--$L^q$ estimates for the spherical maximal function over dilation sets of fractal dimensions, including the endpoint estimates, were recently proved by Anderson--Hughes--Roos--Seeger. More intricate $L^p$--$L^q$ estimates for the fractal circular maximal function were later established in the sharp range by Roos--Seeger, but the endpoint estimates have been left open, particularly when the fractal dimension of the dilation set lies in $[1/2, 1)$. In this work, we prove these missing endpoint estimates for the circular maximal function. We also study the closely related $L^p$--$L^q$ local smoothing estimates for the wave operator over fractal dilation sets, which were recently investigated by Beltran--Roos--Rutar--Seeger and Wheeler. Making use of a bilinear approach, we also extend the range of $p,q$, for which the optimal estimate holds.

math.CA↗
arXiv2025

Weighted Weak Type estimates for non-integral Square Functions

We provide quantitative weighted weak type estimates for non-integral square functions in the critical case $p=2$ in terms of the $A_p$ and reverse Hölder constants associated to the weight. The method of proof uses a decoupling of the role of the weights via a quantitative version of Gehring's lemma. The results can be extended to other $p$ in the range of boundedness of the square function at hand.

math.CA↗
arXiv2025

On differentiation of integrals in Lebesgue spaces

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^ω$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^ω)$ if and only if $p \geq p_0$, thus reproving classical theorems of Hayes in $\mathbb{R}$. The main novelty is that our $\mathcal{B}$ is a Busemann--Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let $\mathcal{B}$ be a basis in a metric measure space $\mathcal{X}$. If $\mathcal{X}$ is complete, then the set $\{ p \in [1,\infty] : \mathcal{B} \text{ differentiates } L^p(\mathcal{X}) \}$ takes one of the six forms\[ \emptyset, \, \{\infty\}, \, [p_0,\infty], \, (p_0,\infty], \, [p_0,\infty), \, (p_0,\infty) \quad \text{for some} \quad p_0 \in [1,\infty). \] Conversely, for every $p_0 \in [1,\infty)$ and each of the six cases above, we construct a complete space $\mathcal{X}$ and a basis $\mathcal{B}$ illustrating the corresponding range of differentiation.

math.CA↗
arXiv2024

Poisson transform and unipotent complex geometry

Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_λ$ which maps generalized functions on the boundary $\partial Z$ to $λ$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N 0})^r<G$ (Iwasawa) and letting the level $a\in A$ tend to $0$ on a ray we retrieve $N$ via $\lim_{a\to 0} Na$ as an open dense orbit in $\partial Z$ (Bruhat). For positive parameters $λ$ the Poisson transform $\mathcal{P}_λ$ is defined an injective for functions $f\in L^2(N)$ and we give a novel characterization of $\mathcal{P}_λ(L^2(N))$ in terms of complex analysis. For that we view eigenfunctions $ϕ= \mathcal{P}_λ(f)$ as families $(ϕ_a)_{a\in A}$ of functions on the $N$-orbits, i.e. $ϕ_a(n)= ϕ(na)$ for $n\in N$. The general theory then tells us that there is a tube domain $\mathcal{T}=N\exp(iΛ)\subset N_\mathbb{C}$ such that each $ϕ_a$ extends to a holomorphic function on the scaled tube $\mathcal{T}_a=N\exp(i\operatorname{Ad}(a)Λ)$. We define a class of $N$-invariant weight functions ${\bf w}_λ$ on the tube $\mathcal{T}$, rescale them for every $a\in A$ to a weight ${\bf w}_{λ, a}$ on $\mathcal{T}_a$, and show that each $ϕ_a$ lies in the $L^2$-weighted Bergman space $\mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a}):=\mathcal{O}(\mathcal{T}_a)\cap L^2(\mathcal{T}_a, {\bf w}_{λ, a})$. The main result of the article then describes $\mathcal{P}_λ(L^2(N))$ as those eigenfunctions $ϕ$ for which $ϕ_a\in \mathcal{B}(\mathcal{T}_a, {\bf w}_{λ, a})$ and $$\|ϕ\|:=\sup_{a\in A} a^{\operatorname{Re}λ-2ρ} \|ϕ_a\|_{\mathcal{B}_{a,λ}}<\infty$$ holds.

math.RT↗
arXiv2024

Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

Let $H_ωf$ be the Fourier restriction of $f\in L^2(\mathbb{R})$ to an interval $ω\subset \mathbb{R}$. If $Ω$ is an arbitrary collection of pairwise disjoint intervals, the square function of $\{H_ωf: ω\in Ω\}$ is termed the Rubio de Francia square function $T^Ω$. This article proves a pointwise bound for $T^Ωf$ by a sparse operator involving local $L^2$-averages. A pointwise bound for the smooth version of $T^Ω$ by a sparse square function is also proved. These pointwise localization principles lead to quantified $L^p(w)$, $p>2$ and weak $L^p(w)$, $p\geq 2$ norm inequalities for $T^Ω$. In particular, the obtained weak $L^p(w)$ norm bounds are new for $p\geq 2$ and sharp for $p>2$. The proofs rely on sparse bounds for abstract balayages of Carleson sequences, local orthogonality and very elementary time-frequency analysis techniques. The paper also contains two results related to the outstanding conjecture that $T^Ω$ is bounded on $L^2(w)$ if and only if $w\in A_1$. The conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue of $T^Ω$.

math.CA↗
arXiv2024

Uniform resolvent estimates, smoothing effects and spectral stability for the Heisenberg sublaplacian

We establish global bounds for solutions to stationary and time-dependent Schrödinger equations associated with the sublaplacian $\mathcal L$ on the Heisenberg group, as well as its pure fractional power $\mathcal L^s$ and conformally invariant fractional power $\mathcal L_s$. The main ingredient is a new abstract uniform weighted resolvent estimate which is proved by using the method of weakly conjugate operators -- a variant of Mourre's commutator method -- and Hardy's type inequalities on the Heisenberg group. As applications, we show Kato-type smoothing effects for the time-dependent Schrödinger equation, and spectral stability of the sublaplacian perturbed by complex-valued decaying potentials satisfying an explicit subordination condition. In the local case $s=1$, we obtain uniform estimates without any symmetry or derivative loss, which improve previous results.

math.AP↗
arXiv2024

Singular integrals along variable codimension one subspaces

This article deals with maximal operators on ${\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional Hörmander--Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1 case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sjölin's generalization of Carleson's maximal operator. Our main result, a weak-type $L^{2}({\mathbb R}^n)$-estimate on band-limited functions, leads to several corollaries. The first is a sharp $L^2({\mathbb R}^n)$ estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson--Sjölin theorem. In addition, we obtain that functions in the Besov space $B_{p,1}^0({\mathbb R}^n)$, $2\le p <\infty$, may be recovered from their averages along a measurable choice of codimension $1$ subspaces, a form of Zygmund's conjecture in general dimension $n$.

math.CA↗
arXiv2024

Landis-type results for discrete equations

We prove Landis-type results for both the semidiscrete heat and the stationary discrete Schrödinger equations. For the semidiscrete heat equation we show that, under the assumption of two-time spatial decay conditions on the solution $u$, then necessarily $u\equiv 0$. For the stationary discrete Schrödinger equation we deduce that, under a vanishing condition at infinity on the solution $u$, then $u\equiv 0$. In order to obtain such results, we demonstrate suitable quantitative upper and lower estimates for the $L^2$-norm of the solution within a spatial lattice $(h\mathbb{Z})^d$. These estimates manifest an interpolation phenomenon between continuum and discrete scales, showing that close-to-continuum and purely discrete regimes are different in nature.

math.AP↗
arXiv2023

Non-existence of radial eigenfunctions for the perturbed Heisenberg sublaplacian

We prove uniform resolvent estimates in weighted $L^2$-spaces for radial solutions of the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proofs are based on the multipliers methods, and strongly rely on the use of suitable multipliers and of the associated Hardy inequalities. The constants in our inequalities are explicit and depend only on the dimension $d$. As application of the method, we obtain some suitable smallness and repulsivity conditions on a complex radial potential $V$ on $\mathbb{H}^d$ such that $\mathcal{L}+V$ has no radial eigenfunctions.

math.AP↗
arXiv2023

Non self-adjoint perturbations of the Heisenberg sublaplacian

We prove uniform resolvent estimates in weighted $L^2$-spaces for the sublaplacian $\mathcal{L}$ on the Heisenberg group $\mathbb{H}^d$. The proof are based on multiplier methods, and strongly rely on the use of horizontal multipliers and the associated Hardy inequalities. The constants of our inequalities are explicit and depend only on the dimension $d$. As applications of this method, we obtain some suitable smallness and repulsivity conditions on a complex potential $V$ on $\mathbb{H}^d$ such that the spectrum of $\mathcal{L}+V$ does not contain eigenvalues.

math.SP↗
arXiv2023

Weak-type maximal function estimates on the infinite-dimensional torus

We prove necessary and sufficient conditions for the weak-$L^p$ boundedness, for $p \in (1,\infty)$, of a maximal operator on the infinite-dimensional torus. In the endpoint case $p=1$ we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

math.CA↗